The 3-restricted edge-connectivity of the direct product graphs

An edge subset $S\subseteq E(G)$ is called a 3-restricted edge-cut if $G-S$ is disconnected and each component of $G-S$ contains at least three vertices. The 3-restricted edge-connectivity of a graph $G$, denoted by ${\lambda }_3(G)$, is defined as the minimum cardinality among all 3-restricted edge-cuts if there are at least one; otherwise, ${\lambda }_3(G)=+\mathrm{\infty }$. It is proved that ${\lambda }_3(G)\le {\xi }_3(G)$ if $G$ has a 3-restricted edge-cut, where ${\xi }_3(G)=\min \{\mathrm{\mid }[X,V(G)\setminus X{]}_G\mathrm{\mid }:\mathrm{\mid }X\mathrm{\mid }=3\text{ and }G[X]\text{ is connected}\}\mathrm{.}$ If ${\lambda }_3(G)={\xi }_3(G)$, then $G$ is said to be maximally 3-restricted edge-connected. The direct product of two graphs $G$ and $H$, denoted by $G\times H$, is defined as the graph with vertex set $V(G\times H)=V(G)\times V(H)$, where two vertices $(u_1,v_1)$ and $(u_2,v_2)$ are adjacent in $G\times H$ if and only if $u_1{u}_2\in E(G)$ and $v_1{v}_2\in E(H)$. In this paper, we determine, for a regular connected graph $G$, the 3-restricted edge-connectivity of $G\times C_n$, $G\times K_n$ and $G\times T_n$, where $C_n$, $K_n$ and $T_n$ are the cycle, the complete graph and the total graph with $n$ vertices, respectively. As corollaries, we establish sufficient conditions for the direct product graphs $G\times C_n$, $G\times K_n$ and $G\times T_n$ to be maximally 3-restricted edge-connected.